1. Chapter 6 Energy and Oscillations
Lecture PowerPoint
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2. Energy and Oscillations
Why does a swinging pendant return to the same point after each swing?

3. Energy and Oscillations
The force does work to move the ball. This increases the ball’s energy, then this energy is used in its motion.

4. Simple Machines, Work, and Power
A simple machine multiplies the effect of an applied force.
For example, a lever :
A small force applied to one end delivers a large force to the rock.
The small force acting through a large distance moves the rock a small distance.

5. Simple Machines, Work, and Power
A simple machine multiplies the effect of an applied force.
For example, a pulley :
Forces balance:
2T = W =>
T= W/2
A small tension applied to one end delivers twice as much tension to lift the box.
The small tension acting through a large distance moves the box a small distance.
Tension in the rope pulls up on either side of the pulley supporting the weight.
Tension = (1/2) weight being lifted, because two ropes pulling up on the pulley.
But move the rope twice the distance that the load moves - both rope segments decrease in length by an amount equal to the increase in the height of the load.
(Force * distance) is same for the input force applied by the person to the rope as for the output force exerted on the load.
Work = Force * distance is conserved (ignoring friction).

6. Simple Machines, Work, and Power
The mechanical advantage of a simple machine is the ratio of the output force to the input force.
For this pulley example, the mechanical advantage is 2.
The output force that lifts the load is twice the input force exerted by the person pulling on the rope.

7. Work is equal to the force applied times the distance moved.
Suppose a constant horizontal force is applied to a heavy crate.
Work is done and the farther it moved, the more work is required.
Work also depends on how hard you push.
If the force and the distance moved are in the same direction, then work is the applied force multiplied by the distance:
Work is equal to the force applied times the distance moved.
Work = Force x Distance: W = F d
Work output = Work input units: 1 Joule (J) = 1 Nm

9. Only forces parallel to the motion do work.
In (a), force on crate was in same direction as motion.
Normal force of the floor pushes upward -- perpendicular to motion => no direct effect in producing the motion.
Forces perpendicular to the motion, such as the normal force or the gravitational force acting on the crate, do no work when the crate moves horizontally.
If the force is neither perpendicular nor parallel to motion, do NOT use the total force in computing work.
Use only force component in the direction of the motion.
Only forces parallel to the motion do work.
In this case, with the block sliding horizontally, only the 30N part of the diagonal force does work.

10. Work is done when a car accelerates, energy: fuel  motion.
How fast? The rate at which work is done depends on the power.
The shorter the time, the greater the power.
If the crate in 6.1(a) is in motion for 10 seconds, the power is found by dividing 200 J by 10 seconds, yielding a power of 20 J/s = 20 W.
A joule per second (J/s) is called a watt (W), the metric unit of power.
OR horsepower (hp). 1hp = 746 watts = kilowatt (kW).

11. A string is used to pull a wooden block across the floor without accelerating the block. The string makes an angle to the horizontal. Does the force applied via the string do work on the block?
Yes, the force F does work.
No, the force F does no work.
Only part of the force F does work.
You can’t tell from this diagram.
Only the part of the force that is parallel to the distance moved does work on the block. This is the horizontal part of the force F.

12. If there is a frictional force opposing the motion of the block, does this frictional force do work on the block?
Yes, the frictional force does work.
No, the frictional force does no work.
Only part of the frictional force does work.
You can’t tell from this diagram.
Since the frictional force is antiparallel to the distance moved, it does negative work on the block.

13. Does the normal force of the floor pushing upward on the block do any work?
Yes, the normal force does work.
No, the normal force does no work.
Only part of the normal force does work.
You can’t tell from this diagram.
Since the normal force is perpendicular to the distance moved, it does no work on the block.

14. A force of 50 N is used to drag a crate 4 m across a floor
A force of 50 N is used to drag a crate 4 m across a floor. The force is directed at an angle upward from the crate as shown. What is the work done by the horizontal component of the force?
120 J
160 J
200 J
280 J
0 J
The horizontal component of force is 40 N and is in the direction of motion.
W = F · d
= (40 N) · (4 m)
= 160 J.

15. What is the work done by the vertical component of the force?
120 J
160 J
200 J
280 J
0 J
The vertical component of force is 30 N but isn’t in the direction of motion:
W = F · d
= (30 N) · (0 m)
= 0 J.

16. What is the total work done by the 50-N force?
120 J
160 J
200 J
280 J
0 J
Only the component of force in the direction of motion does work:
W = F · d
= (40 N) · (4 m)
= 160 J.

17. Kinetic Energy
Suppose force applied to move a crate is the only force acting on the crate in the direction of motion.
Newton’s 2nd law => crate accelerates => velocity increases.
Doing work on an object increases its energy.
Energy associated with the motion of the object is kinetic energy.
Work transfers energy => kinetic energy gained = work done.
Assume, crate is on rollers => friction may be small enough to be ignored.
Force accelerates the crate. Acceleration = Force/mass. Crate gains speed, move faster
Chapter 2=> for constant acceleration distance proportional to square of final speed. => Work done is also proportional to the square of the speed.
Work done = increase in kinetic energy =>
kinetic energy must increase with the square of the speed.
Exact formula derivation requires Calculus.
If the crate is initially at rest, its kinetic energy is equal to zero. After being accelerated over a distance d, it has a final kinetic energy = mv^2/2 = work done on the crate.
The work done is actually equal to the change in kinetic energy. If the crate was already moving when you began pushing, its increase in kinetic energy would equal the work done.

18. Calculate the energy gained by the crate in two different ways.
In the first method, we use the definition of work.
In the second, we use the definition of kinetic energy. We find that 200 J of work done on the crate results in an increase in kinetic energy of 200 J. It is no accident that these values are equal. Our definition of kinetic energy guarantees this to be true.

19. W = −fd, Forces can decelerate objects as well as accelerate them.
Apply car brakes and it skids to a stop – looses kinetic energy.
Decrease in kinetic energy = negative change in kinetic energy.
=> Work done by frictional forces on the car by is also negative.
Frictional force exerted on the car is opposite to motion.
It does negative work, removing energy from the system.
For a frictional force of magnitude f, the work done is:
W = −fd,
if the car moves a distance d while decelerating.
Frictional forces always oppose the direction of motion=>
Work done by friction is always negative.

20. Stopping distance for a moving car
The kinetic energy of the car is proportional speed squared.
Double the speed, the kinetic energy quadruples.
Four times as much work must be done to reach the doubled speed.
Likewise, four times as much energy must be removed to stop the car.
A practical application is the stopping distances of cars.
Negative work required to stop the car = kinetic energy (removed). Work required (and stopping distance) increases rapidly with speed.
Doubling speed (3060)requires four times as much negative work.
Work done is proportional to the distance.
=>Stopping distance at 60 MPH = four times that required at 30 MPH. In reality, the frictional force varies with the speed of the car.
If you look at the stopping distances in driver-training manuals, you will see that they do indeed increase rapidly with speed, although not exactly in proportion to the square of the speed.
The more kinetic energy present initially, the more negative work is required to reduce this energy to zero => greater the stopping distance.

21. Potential Energy
Lift a crate to a higher position on a loading dock.
Work is done, but no kinetic energy is gained if the crate ends up just sitting on the dock.
Similarly, drawing back a bowstring or compressing a spring. Work is done, but no kinetic energy is gained.
If work is done but no kinetic energy is gained, we say that the potential energy has increased.
Gravitational potential energy
The applied force is NOT the only force acting on the crate.
Gravitational force(weight) pulls down on the crate.
If lifting force = gravity => net force = zero => acceleration = 0.
Work done by lifting force increases the gravitational potential energy of the crate. Work is done by moving against the gravitational pull.
Work of the lifting force = force * distance moved.
Applied force = weight of the crate = mg => work = mgh
The change in gravitational potential energy is equal to the work done:
Potential Energy (PE) = mgh (h=distance moved compared to reference level 0)
We usually choose the lowest point in the probable motion of the object as the reference level to avoid negative values of potential energy. The changes in potential energy are what is important, however, so the choice of reference level does not affect the physics of the situation.

22. The essence of potential energy
Potential energy -- storing energy to use later on for other purposes.
The crate could be left indefinitely higher up on the loading dock.
If we pushed it off, it would rapidly gain kinetic energy as it fell.
The kinetic energy, in turn, could be used to compress objects underneath, to drive pilings into the ground, etc… (fig. 6.10).
Kinetic energy also has this feature, however, so storing energy is not what distinguishes potential energy.
Potential energy involves changing position of the object acted on by a specific force.
For gravitational potential energy, force = gravitational attraction of the Earth.
The farther we move the object away from the Earth, the greater the gravitational potential energy. Other kinds of potential energy involve different forces.

23. Work is done on a large crate to tilt the crate so that it is balanced on one edge, rather than sitting squarely on the floor as it was at first. Has the potential energy of the crate increased?
Yes No
Yes. The center of the crate has been lifted slightly. If it is released it will fall back and convert the potential energy into kinetic energy.

24. Elastic potential energy
For pulled bowstring or stretched a spring, work is done by applied force against an opposing elastic force - results from stretching/compressing.
If pull block from the original (equilibrium, un-stretched) position, the system gains elastic potential energy.
If let go, the block would fly back.
Force is applied over some distance to move the block =>
Work is done in pulling against the force exerted by the spring.
Most springs force proportional to stretched distance x from equilibrium:
, where k = spring constant - describes spring’s stiffness.
-- Hooke’s Law (R. Hooke 1635–1703).
The minus sign indicates that the force exerted by the spring pulls back on the object as the object moves away from its equilibrium position. Thus, if the mass is moved to the right, the spring pulls back to the left.
If the spring is compressed, it pushes back to the right.

25. Elastic potential energy
Increase in potential energy of elastic system = work done by the force.
Want block to move without acceleration => net force on it = 0.
=> applied force must be adjusted so that it is always equal in magnitude but opposite in direction to the force exerted by the spring.
=> applied force must increase as the distance x increases (fig. 6.12).
The increase in elastic potential energy = work done by the average force needed to stretch the spring.
Fig => average force = final force /2 =
=> work done is the average force times the distance x =
=> Potential energy of the stretched-spring system =
The same expression is valid when the spring is compressed. The distance x is then the distance that the spring is compressed from its original relaxed (equilibrium) position.
The potential energy stored in the spring can be converted to other forms. If we let go of the block when the spring is either stretched or compressed, the block will gain kinetic energy. Cocking a bow and arrow, squeezing a rubber ball, and stretching a rubber band are all elastic energy examples.

26. Conservative Forces
Potential energy results from work done against a variety of different forces besides gravity and springs.
Work done against frictional forces, however, does NOT result in an increase in the potential energy of the system.
Instead, heat is generated, which either transfers energy out of the system or increases the internal energy of the system at the atomic level.
As discussed in chapter 11, this internal energy cannot be completely recovered to do work on another object or system.
Forces such as gravity or elastic forces that lead to potential energy relationships are referred to as conservative forces.
When work is done against conservative forces, the energy gained by the system is completely recoverable for use in other forms.

27. Conservation of Energy:
Total energy = kinetic + potential energies is conserved in many situations.
Consider a pendulum; pull the ball to the side and release it to start it swinging.
1st work is done by you to increase potential energy of the ball = mgh
It becomes gravitational potential energy, PE = mgh, where h = height above its initial. Release the ball, potential energy is transferred to kinetic energy as it swings.
At the bottom of the swing (the initial position of the ball when it was just hanging), the potential energy is zero, and the kinetic energy reaches its maximum value.
It doesn’t stop at low point; it continues to a point opposite the release point.
Kinetic energy decreases while potential energy increases until it reaches the point where the kinetic energy is zero and the potential energy is equal to its initial.
Then repeat the motion.

28. What does it mean to say that energy is conserved?
Pendulum swings -- continuing change of potential energy to kinetic energy and.
The total mechanical energy = potential + kinetic = const because there is no work being done on the system to increase or decrease its energy.
When only conservative forces are involved, the total mechanical energy of the system (the sum of its kinetic energy and potential energy) remains constant.
Work is pivotal. If no energy is added or removed by forces doing work, the total energy should not change: If W = 0, E = PE + KE = constant
Note that gravitational force becomes part of the system by including the gravitational potential energy of the ball in our description.
Gravity is a conservative force already accounted for by potential energy.
Tension of the string force is perpendicular to the motion => no work!!!
No component in the direction of motion.
Air resistance force does negative work on the ball, slowly decreasing the total mechanical energy of the system.
The total energy of the system is not completely constant.
It would be constant only if air resistance were negligible.

29. Why do we use the concept of energy?
There are advantages of using the principle of conservation of energy.
It is harder to describe motion of pendulum using only Newton’s 2nd law.
Forces vary continually in direction and magnitude as the pendulum moves.
Energy approach allows to make predictions much easier
Ignoring friction, predict that the ball will reach the same height at either end of swing. The kinetic energy is zero at the end points of the swing where the ball momentarily stops, and at these points, the total energy equals potential energy.
If no energy has been lost, the potential energy has the same value that it had at the point of release, which implies that the same height is reached (PE = mgh).
Demonstration: big metal ball is suspended at ceiling near the chin of the physics instructor.
Releases it to allow it to swing and return just a few inches from her chin (fig. 6.15).
Note that no initial velocity must be given or it may go higher!!!!

30. Also conservation of energy can be used to predict the speed at any point in the swing.
It is zero at the end points and has its maximum value at the low point of the swing.
Put reference level for potential energy at low point =>
potential energy =0 there because the height is zero.
All of the initial potential energy has been converted to kinetic energy => find speed.
We could find the speed at any other point in the swing by setting the total energy at any point equal to the initial energy.
Different values of the height h above the low point yield different values of the potential energy. The remaining energy must be kinetic energy.
The system has only so much energy, either potential or kinetic energy or some of both, but it cannot exceed the initial value.

31. How is energy analysis like accounting?
A sled on a hill and a roller coaster illustrate the principle of conservation of energy. Conservation of energy can predict speed! Very hard with Newton’s laws.
An energy accounting provides a better overview.
A parent pulls the sled to the top of a hill, doing work that increases potential energy.
Parent may do more work by giving the sled a push -- initial kinetic energy.
Total work by parent = energy input to the system = potential + kinetic energies.
This initial energy came from the body of the parent doing the pulling and pushing.. That energy came from food, which in turn involved solar energy stored by plants.

32. If negligible friction and air resistance, energy is conserved:
total energy at any point = the initial energy.
It is more realistic to assume that there is some friction.
Hard to predict work done against friction, only estimate - assumed 2000 J total.
The work done against friction removes energy from the system -- expenditure.
The energy balance at the bottom of the hill is 8200 J, rather than J.
This will lead to a smaller, more realistic speed at the bottom.
Any work done by frictional forces is negative.
That work removes mechanical energy from the system.

33. A sled and rider with a total mass of 40 kg are perched at the top of the hill shown. Suppose that 2000 J of work is done against friction as the sled travels from the top (at 40 m) to the second hump (at 30 m). Will the sled make it to the top of the second hump if no kinetic energy is given to the sled at the start of its motion?
yes no
It depends.
Yes. The difference between the potential energy at the first point and the second point, plus loss to friction is less than the kinetic energy given at the start of the motion.

34. A lever is used to lift a rock
A lever is used to lift a rock. Will the work done by the person on the lever be greater than, less than, or equal to the work done by the lever on the rock?
Greater than
Less than
Equal to
Unable to tell
from this diagram
The work done by the person can never be less than the work done by the lever on the rock. If there are no dissipative forces they will be equal. This is a consequence of the conservation of energy.

35. Springs and Simple Harmonic Motion
Many systems involve springs or elastic bands that move back and forth.
Potential energy being converted to kinetic energy and then back.
Mass on spring and simple pendulum -- simple harmonic motions where energy of a system repeatedly changes from potential energy to kinetic energy and back again.
Pull it to one side of its equilibrium position (where spring not stretched/compressed). Doing work to pull the mass against the opposing force of the spring increases the potential energy of the spring-mass system = similar to bow or slingshot.
Release mass, potential energy is converted to kinetic energy.
Like pendulum mass goes beyond equilibrium
and spring compresses, gaining potential energy.
When the kinetic energy is completely reconverted to potential energy, the mass stops and reverses, and the whole process repeats.
Energy changes continually from potential energy to kinetic energy and back again.
If friction can be ignored, total energy = const, and mass oscillates back and forth.
Using a video camera or other tracking techniques, it is possible to measure and plot the position of a pendulum bob.

37. Plot the position of a pendulum bob or mass on a spring vs. time.
It is harmonic function (sin/cos) -- simple harmonic motion (musical)
Sounds produced by vibrating strings, reeds, and air columns
The line at zero is the equilibrium position for the mass on a spring.
Points above this line represent positions on one side of the equilibrium point, and those below the line represent positions on the other side.
The motion starts at the point of release, where the distance of the mass from equilibrium is a maximum.
As the mass moves toward the equilibrium position (x = 0 on the graph) it gains speed, indicated by the increasing slope of the curve.
The object’s position changes most rapidly when it is near the equilibrium point, where the kinetic energy and speed are the greatest.
Mass passes through equilibrium position, then moves away opposite to its initial position.
The force exerted by spring is now in opposite to the velocity and is decelerating the mass.
When the mass reaches the point farthest from its release point, the speed and kinetic energy are again zero, and the potential energy has returned to its maximum value.
The slope of the curve is zero at this point, indicating that the mass is
momentarily stopped (its velocity is zero).

38. What are the period and the frequency?
Curve repeats itself regularly.
The period T is the repeat time = time for one complete cycle.
You can think of the period as the time between adjacent
peaks or valleys on the curve.
A slowly oscillating system has a long period, rapidly oscillating system -- short period.
Let T = 0.5 sec => two oscillations each second = f = frequency of oscillation.
Frequency f is the number of cycles per unit time = f = 1/T. (1Hz – once cycle per sec )
A rapidly oscillating system has a very short period and thus a high frequency.
Loose spring generally has low frequency of oscillation and a stiff spring to have a high. Larger masses offer greater resistance to motion, producing lower frequencies.
The period and frequency of oscillation of a pendulum depend primarily on its length, measured from the pivot point to the center of the bob.
To measure the period, you usually measure the time required for several complete swings and then divide by the number of swings to get the time for one swing.

39. Will any restoring force produce simple harmonic motion
Will any restoring force produce simple harmonic motion? When a mass on spring is moved off equilibrium, the spring exerts a force that pulls or pushes the mass back toward the center -- restoring force.
In any oscillation requires such force.
Spring force is proportional to the distance x from equilibrium position (F = −kx).
Simple harmonic motion results for such restoring forces.
If restoring force is complicated, may get oscillation but not simple harmonic motion.
Easiest spring-mass system is suspended from ceiling – no friction.
Oscillates up and down rather than horizontally.
Two forces - spring force pulling upward and the gravitational force pulling downward. Gravitational force = constant => it just moves the equilibrium point lower.
The equilibrium point is where the net force is zero—the downward
pull of gravity is balanced by the upward pull of the spring.
The variations in the restoring force are still provided by the spring.
These variations are proportional to the distance from equilibrium
just as they are in the horizontal case – also simple harmonic motion.
BUT, potential energy gravitational + elastic

40. For simple pendulum - gravity is the restoring force.
Away from equilibrium, gravity pulls it back.
The part of the gravitational force in the direction of motion is proportional to the displacement (as long as it is not too large).
=> For small amplitudes get simple harmonic motion.